A STUDY OF TEACHER TRANSITIONS TO A REFORM-BASED MATHEMATICS CURRICULUM IN AN URBAN SCHOOL: THE INTERACTION OF BELIEFS,
KNOWLEDGE, AND CLASSROOM PRACTICES
Wendy S. Bray
A dissertation submitted to the faculty of the University of North Carolina at Chapel Hill in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the School of Education (Culture, Curriculum, and Change).
Chapel Hill 2007
Approved by
ABSTRACT
Wendy S. Bray: A Study of Teacher Transitions to a Reform-Based Mathematics Curriculum in an Urban School: The Interaction of Beliefs, Knowledge, and Classroom Practices
(Under the direction of Mary Ruth Coleman)
This collective case study examines how four third-grade teachers’ beliefs and knowledge influenced their ways of supporting and limiting student thinking in their first year using a reform-based mathematics curriculum at an urban school. Of focus is the role teachers’ beliefs and knowledge play in supporting and limiting student thinking when student difficulties arise during instruction on multiplication and division. Situated in the growing body of research associated with current reforms in mathematics education, this study is also informed by general education research on urban schools, teacher beliefs, teacher knowledge, and teacher change.
Findings from this study suggest that some aspects of reform-oriented mathematics instruction are more readily adopted than others. While beliefs and knowledge both appear to influence teacher response to student difficulties, certain aspects of instruction seem more greatly influenced by teacher beliefs while others appear more greatly influenced by teacher knowledge. In addition, evidence suggests that teachers’ differential classroom experiences during initial use of reform-based mathematics curriculum were related to the degree to which teachers’ evolving beliefs and knowledge moved closer to alignment with reform-based mathematics practices. Finally, the urban context of this study was found to influence teachers’ transitions to reform-based mathematics teaching practices in a variety of ways.
Study findings have several implications for efforts to support teachers’ transitions to reform-based mathematics programs and practices within and outside of urban school
ACKNOWLEDGEMENTS
I am indebted to all of the individuals who participated in this study. I especially want to acknowledge the generosity and courage of the teachers who welcomed me into their classrooms to study their initial efforts at reform-based mathematics teaching.
I wish to thank my dissertation committee – Drs. Mary Ruth Coleman, Juli Dixon, Susan Friel, Judith Meece, and Dwight Rogers – for their thoughtful advice and
encouragement during this research project and throughout my doctorial studies. I am especially grateful to my dissertation chair, Dr. Mary Ruth Coleman, for all of her support, both personal and professional.
TABLE OF CONTENTS
LIST OF TABLES ... xi
LIST OF FIGURES ... xiii
Chapter I. INTRODUCTION ... 1
Overview of Study ... 1
Purpose of Study and Research Questions...3
Focus on Teacher Response to Student Difficulties during Problem Solving...4
Review of the Literature ... 5
Mathematics Reform and Student Thinking ...5
The Challenge of Reforming Mathematics Teaching...12
The Challenge of Urbanicity...15
Knowledge for Teaching Mathematics ...17
Knowledge for Teaching Multiplication and Division ...25
Beliefs and Mathematics Teaching...33
Beliefs Beyond Mathematics ...40
The Process of Reforming Mathematics Teaching...43
A Synthesis of the Research Literature – My Analytical Framework ... 48
II. RESEARCH METHOD... 52
Research Design... 53
Overview and Justification of Research Design ...53
Establishing Transparency of Researcher Position...56
Sample Selection Procedures ... 58
Selection of the Research Site...59
Selection of Case Study Participants ...60
Selection of School Leader Participants ...61
Data Collection Procedures... 62
Core Classroom Observations...62
Pre- and Post-Observation Interviews ...68
Beliefs Survey...71
Teacher Knowledge Interview...73
Records of the Teacher Development Project ...75
Interviews with School Leaders...76
Student Data...78
Data Management Procedures ... 78
Data Analysis Procedures ... 80
Analysis during Data Collection...80
Analysis after Data Collection ...81
Development of Case Stories...89
Cross-case Analysis ...90
III.FINDINGS ... 93
The School Context: Lincoln Heights Elementary ... 94
School Culture, History, and Demographics ...94
Mathematics-Related New Initiatives...97
Mathematics Teacher Development Project ...101
The Case Study Teachers and Their Classes ... 106
Ms. Aria and her Class...108
Ms. Jarmin and her Class ...109
Ms. Larsano and her Class ...110
Ms. Rosena and her Class ...111
The Case of Ms. Aria ... 113
Ms. Aria’s Beliefs about Mathematics Teaching and Learning ...114
Ms. Aria’s Knowledge of Mathematics for Teaching ...121
Case Story of Ms. Aria’s Response to Student Difficulties...131
Situating Ms. Aria’s Case Story in Broader Measures of Teaching ...150
The Case of Ms. Jarmin ... 157
Ms. Jarmin’s Beliefs about Mathematics Teaching and Learning...159
Ms. Jarmin’s Knowledge of Mathematics for Teaching...167
Case Story of Ms. Jarmin’s Response to Student Difficulties ...176
Situating Ms. Jarmin’s Case Story in Broader Measures of Teaching ...197
The Case of Ms. Larsano ... 203
Ms. Larsano’s Beliefs about Mathematics Teaching and Learning...205
Ms. Larsano’s Knowledge of Mathematics for Teaching...211
Case Story of Ms. Larsano’s Response to Student Difficulties ...222
Situating Ms. Larsano’s Case Story in Broader Measures of Teaching ...242
The Case of Ms. Rosena ... 248
Ms. Rosena’s Beliefs about Mathematics Teaching and Learning...250
Ms. Rosena’s Knowledge of Mathematics for Teaching...258
Case Story of Ms. Rosena’s Response to Student Difficulties ...268
IV.CROSS-CASE ANALYSIS & DISCUSSION... 296
Discussion of Findings across Cases ...297
Teacher Actions that Support and Limit Student Thinking ...297
The Influence of Beliefs and Knowledge on Teachers’ Response to Student Difficulties ...301
Relationship among Teachers’ Evolving Beliefs, Knowledge, and Mathematics Instruction...313
The Influence of the Urban School Context ...318
Study Implications ...325
Study Limitations and Directions for Future Research...330
Conclusions...333
LIST OF TABLES
Table
1. Equal-Grouping Problem Types ...26
2. Composition of the Student Population at Lincoln Heights and the School District ...59
3. Characteristics of Case Study Teacher Participants ...61
4. Relationship Between Research Questions and Data Collection ...63
5. Protocol for Core Classroom Observations ...64
6. Overview of Observed Lessons ...65
7. Sample Questions from Interviews with School and School-District Leaders...77
8. Sample of Codes Used for Teacher Actions that Support and Limit Student Thinking ...83
9. Codes Used to Identify Occurrence of Student Difficulty...85
10. Focus of Third-grade Workshops ...102
11. Student Demographics of Case Study Teachers’ Classes...107
12. Ms. Aria’s IMAP Web-Based Beliefs-Survey Results...114
13. Strategies Identified by Ms. Aria in Response to Classroom Scenario 1 ...125
14. Ratings of Ms. Aria’s Mathematics Teaching on the RTOP ...153
15. Comparison of Aria Student Achievement on SST to School, District, and State ...156
16. Ms. Jarmin’s IMAP Web-Based Beliefs-Survey Results ...159
17. Strategies Identified by Ms. Jarmin in Response to Classroom Scenario 1 ...170
18. Ratings of Ms. Jarmin’s Mathematics Teaching on the RTOP ...199
19. Comparison of Jarmin Student Achievement on SST to School, District, and State...202
21. Strategies Identified by Ms. Larsano in Response to Classroom Scenario 1 ...214
22. Ratings of Ms. Larsano’s Mathematics Teaching on the RTOP ...244
23. Comparison of Larsano Student Achievement on SST to School, District, and State...247
24. Ms. Rosena’s IMAP Web-Based Beliefs-Survey Results ...251
25. Strategies Identified by Ms. Rosena in Response to Classroom Scenario 1 ...262
26. Ratings of Ms. Rosena’s Mathematics Teaching on the RTOP ...290
27. Comparison of Rosena Student Achievement on SST to School, District, and State...294
28. Summary of Teaching Strategies Observed to Support and Limit Student Thinking...298
29. Summary of the Relationship Among Teacher Response to Student Difficulties, Teacher Beliefs, Teacher Knowledge, and Student Thinking...302
LIST OF FIGURES
Figure
1. Naming Conventions for Arrays ...30
2. An Interactive Perspective of Teachers' Knowledge, Beliefs and Experiences...49
3. Pre-Observation Interview Protocol ...69
4. Post-Observation Interview Protocol ...70
5. Seven Teacher Beliefs Measured by the IMAP Survey...71
6. Summary of Knowledge Interview Protocol ...74
CHAPTER I
INTRODUCTION
Overview of Study
The reform agenda in mathematics education, exemplified by the National Council of Teachers of Mathematics (NCTM) standards documents (1989, 2000), calls for significant changes in the ways teachers orchestrate mathematics learning for students. The NCTM standards contend that elementary school mathematics should emphasize problem-solving, reasoning, and communicating mathematical ideas, which places student thinking and
conceptual understanding at the center of the instructional agenda. Teachers are challenged to develop discourse-rich mathematics learning communities where students devise, share, and analyze multiple solution strategies in response to rich mathematics tasks. The new forms of teaching and learning required to meet these standards place significant demands on teachers. Not only are new kinds of mathematics and pedagogical knowledge required (Ball,
center on the dramatic changes to teachers’ mathematics instructional practices envisioned by these policies.
Research on teacher change suggests that teachers’ beliefs about and knowledge of subject-matter, teaching, and learning influence the ways they revise their teaching practices in response to reform recommendations (Richardson & Placier, 2001; Spillane & Jennings, 1997). If a teacher’s beliefs are in opposition to the pedagogical emphasis of mathematics reform or if teachers do not have adequate knowledge to support reform-based mathematics teaching practices, then the success of policy initiatives will be limited. This has strong implications for the ways reform policies are implemented, particularly at schools serving high numbers of children from poverty where teachers traditionally rely on controlled classroom environments to build basic skills (Anyon, 1981; Knapp, 1995a). Even though the exclusive focus on basic skills is inconsistent with the vision promoted by NCTM, many urban school teachers believe strongly that students’ needs demand it. A limited number of professional development projects have been successful in helping urban school teachers to reform their teaching practices with positive outcomes for students (Campbell, 1996; Fuson, Smith, & Lo Cicero, 1997; Hufferd-Ackles, Fuson, & Sherin, 2004; Knapp, 1995b). The question is what more can be done to support teacher transition to a reform-based
among beliefs, knowledge, and practice appears to be an interactive one in which teacher trajectories of change take a variety of paths. Research is needed to explore the interactive nature of this relationship as well as the developmental patterns between all three. Such research could inform efforts to help teachers that differ in their beliefs, knowledge and practice become comfortable following the path of reform-based mathematics in any school setting including challenging urban schools.
Purpose of Study and Research Questions
The purpose of this study is to examine how urban school teachers’ beliefs and knowledge influence the ways they support and limit student thinking during their first year using a reform-based mathematics curriculum. The school identified for this study is located in an urban setting and serves a student population characterized by high-poverty and limited English proficiency. By carefully studying a few urban school teachers during a time of change, this study aims to contribute to the growing knowledge of how teachers can be supported in their efforts to transition to reform-oriented mathematics pedagogy, especially in similar urban school settings.
The following broad questions guided this study:
1. In what ways and to what extent do teachers support and limit student thinking during mathematics instruction in their first year implementing a reform-based mathematics curriculum?
3. How does teacher knowledge of mathematics for teaching influence the ways teachers incorporate student thinking in their first year of implementing a reform-based mathematics curriculum?
4. How does the urban context, as defined by the research literature and perceived by teachers and school leaders, influence mathematics instruction in this urban school? These questions are addressed using data from a collective case study of four third-grade urban school teachers during their first year implementing a reform-based mathematics program.
Focus on Teacher Response to Student Difficulties during Problem Solving
likely. Therefore, study of teachers’ response to student difficulties is especially relevant to urban schools.
Review of the Literature
This study is informed by the growing body of research associated with current reforms in mathematics education as well as research on urban schools, teacher beliefs, and teacher knowledge. This section will begin by describing the vision of mathematics teaching and learning promoted by mathematics-reform with special attention to the role of student thinking. Then the challenges teachers face in revising their classroom practices to reflect this vision will be discussed, followed by a review of additional challenges associated with urban schools. Next, a review of research on teachers’ knowledge and beliefs will be provided with focus on their relationship to mathematics teaching practices. Within this discussion,
knowledge for teaching multiplication and division will be described, as these are the mathematics topics of focus during classroom observations for this study. Finally, research on teacher change to reform-based mathematics pedagogy will be reviewed.
Mathematics Reform and Student Thinking
Beginning in the late 1980s, the National Council of Teachers of Mathematics
The need for curricular reform in K-4 mathematics is clear. Such reform must address both the content and emphasis of the curriculum as well as approaches to instruction. A long-standing preoccupation with computations and other traditional skills has dominated both what mathematics is taught and the way mathematics is taught at this level. As a result, the present K-4 curriculum is narrow in scope; fails to foster
mathematical insight, reasoning, and problem solving; and emphasizes rote activities. Even more significant is that children begin to lose their belief that learning
mathematics is a sense-making experience. They become passive receivers of rules and procedures rather than active participants in creating knowledge (p.15).
In this initial standards document and in a more recent document titled Principles and
Standards for School Mathematics (2000), NCTM advocates for mathematics curriculum and instruction that emphasizes the interwoven development of conceptual understanding and procedural knowledge. This approach places problem solving and reasoning at the center of the mathematics classroom experience. It also suggests significantly different roles for students and teachers.
Reflecting constructivist views of learning, advocates of reform-based mathematics instruction believe that children actively construct increasingly organized structures of knowledge and personal understanding by reflecting on and reasoning about experiences in relation to their prior knowledge and immediate contexts (von Glasersfeld, 1996). Therefore, students’ differential realities and existing knowledge constructions become the starting point for conceptually-focused instruction (Wood, Cobb, & Yackel, 1995). Children develop increasingly sophisticated understandings by constructing relationships among mathematical ideas, extending and applying mathematical knowledge, reflecting on experiences,
articulating and defending their thinking, and by making mathematical knowledge their own (Carpenter & Lehrer, 1999).
mathematics instructional practices. In general, such classrooms are places where,
“…students are encouraged to be curious about mathematical ideas, where they can develop their mathematical intuition and analytical capabilities, where they learn to talk about and with mathematical expertise" (Franke, Kazemi, & Battey, 2007). Several research efforts have illuminated classroom efforts to make students’ mathematical thinking a central feature of instruction (Fraivillig, Murphy, & Fuson, 1999; Franke et al., 1997; Hiebert et al., 1997; Kazemi & Stipek, 2001; Stigler, Fernandez, & Yoshida, 1996).
In all classrooms, teachers are the gatekeepers to the kinds of tasks and activities students encounter. Tasks that support student thinking are intentionally selected or designed with specific mathematical goals in mind, and they are viewed as opportunities for students to grapple with mathematics as problem solvers, not rule-followers (Hiebert et al., 1997). In reform-oriented mathematics classrooms, tasks are often situated in real-world contexts that allow students to use their existing knowledge to explore mathematical ideas before they are formally introduced (Carpenter, Fennema, & Franke, 1996). In these classrooms, teachers support student thinking by providing ample opportunity for students to solve mathematics-rich tasks in their own ways (Franke et al., 1997; Stigler et al., 1996) with access to a variety of tools and resources to support mathematical thinking (Hiebert et al., 1997).
emphasize discussion, collaboration, and negotiation as ways of fostering shared meaning among a community of learners (Cobb, Boufi, McClain, & Whitenack, 1997; Gergen, 1995). Students are expected to seek and understand relations among multiple ways of solving mathematical problems, and the classroom discourse is consistently focused on this goal. Students’ flawed solution strategies are readily incorporated into class discussion in order to explore the contradictions in student solutions and provide greater insight into the
mathematics of focus (Kazemi & Stipek, 2001).
Sociomathematical norms have been introduced as important mathematics-specific norms that constitute what counts as mathematical thinking in the classroom (Yackel & Cobb, 1996). Kazemi and Stipek (2001) have identified a pattern of sociomathematical norms in classrooms that are associated with a high level of mathematical understanding. In these classrooms, it is expected that students explain how they solve problems by providing mathematical arguments along with procedural descriptions. Students are expected to seek and understand relations among multiple strategies, and the classroom discourse is
consistently focused on this goal. When students work with a partner or a group, all students are individually accountable for understanding, and they are expected to reach consensus through mathematical argumentation. Finally, when solutions are presented that contain errors, the sociomathematical norm of classroom practice is to explore the contradictions in student solutions and use the error as an opportunity to rethink the problem.
compared to a strategy reporting classroom culture. In both kinds of classrooms, students solve problems in a variety of ways and significant instructional time is dedicated to class discussion of students’ varying mathematical solutions. In the strategy reporting classroom culture, however, interactions during class discussions were primarily limited to interaction between the individual students sharing and the teacher, limiting the involvement and
learning of the rest of the class. In contrast, the inquiry/argument classroom culture involved class members in asking questions and making judgments about the reasonableness of a method when their peers are presenting mathematical solutions. This greater level of minds-on participatiminds-on in the classroom discourse led to more sophisticated student thinking. This study illustrates the importance of the teacher’s role in facilitating classroom discourse in ways that simultaneously support the learning of students who are sharing their mathematical ideas as well as the other students in the class (Franke et al., 2007).
Other research studies have found positive correlations between factors associated with an instructional emphasis on student thinking and more general measures of
mathematics student achievement (Franke et al., 2007). Fennema and her colleagues (1996) conducted a three-year study of first-grade teachers who were attempting to develop teaching practices intentionally grounded in knowledge of student thinking. This study concluded that instructional practices most associated with higher student achievement include a focus on problem-solving, ample opportunity for students to engage in conversation about
Clearly, instruction focused on student thinking changes the nature of the teacher’s role in the classroom. Fraivillig, Murphy, and Fuson (1999) have devised a pedagogical framework elaborating the teacher’s role in advancing children’s mathematical thinking through classroom interaction. The three overlapping components comprised in this framework include the teacher’s role in eliciting children’s solution methods, supporting children’s conceptual understanding, and extending children’s mathematical thinking. The first component, eliciting children’s solution methods, focuses on the strategies teachers use to give students opportunity and encouragement to express their mathematical thinking. Other research has also highlighted the importance of teachers eliciting student thinking in informal and group interactions (Franke et al., 1997; Stigler et al., 1996). Strategies used to elicit student thinking provide a window for the teacher into understanding how children are thinking about mathematics. By skillfully orchestrating class discussions focused on
students’ many ways to solve a problem, the teacher can simultaneously facilitate group learning and engage in assessment of the thinking of individuals and the group.
might record students’ solution methods on the board as a way of scaffolding the problem solver explaining her solution while at the same time aiming to support the other students who are trying to make sense of the solution. If the problem solver hits a sticking point, the teacher may ask the class to help the student work through it. This action primarily supports the thinking of the student describing a solution, but it also supports the other students in maintaining engagement and considering the problem from the perspective of the given solution.
While the eliciting and supporting components of the Fraivillig et al. (1999) framework elaborate how teachers might facilitate children’s understanding of familiar solution methods, the final component focuses on strategies that aim to challenge and extend children’s current mathematical thinking. Strategies that aim to extend children’s thinking encourage students to press beyond their initial solutions to understand alternative solutions. Students are encouraged to analyze and reflect on patterns, draw generalizations across various student strategies, and to consider relationships among mathematical concepts. This component also includes some strategies that aim to influence students’ dispositions toward mathematics, including fostering perseverance and a love of challenge.
The Challenge of Reforming Mathematics Teaching
National reform efforts in mathematics education have had widespread impact on general education curriculum policy, with most states revising state curriculum documents to align with the NCTM standards (Pugach & Warger, 1996). Yet many teachers, the front-line implementers of reform-ideas, have not made the dramatic changes to their mathematics instructional practices envisioned by these policies (Spillane & Jennings, 1997). Some would say that this is because teachers are resistant to change (McLaughlin, 1987). However, an alternative view that is increasingly accepted is that teaching is a more complex activity than has been historically acknowledged (Ernest, 1989; Richardson & Placier, 2001). This seems especially true of the current mathematics-reform efforts that call for teachers to emphasize student thinking.
Sherin (2002a) compares facilitating classroom discussion of student ideas to a balancing act. She describes the constant tension between attempting to support student-centered processes of classroom discourse while, at the same time, facilitating discussion of significant mathematics content. These kinds of discussions also challenge teachers to facilitate classroom discourse such that students do the majority of the intellectual work of unpacking mathematical ideas such that they are comprehensible to classmates (Franke et al., 2007). For the teacher, this involves cultivating questioning techniques that focus students’ attention on important mathematical ideas while at the same time being careful to avoid funneling the conversation such that the teacher takes on the majority of the intellectual work (Wood, 1998). At the same time, the teacher is responsible for facilitating class discussions such that all students participate in active and productive ways (Williams & Baxter, 1996). These tensions make teaching mathematics more like improvisation than a choreographed dance (Heaton, 2000).
Teachers accustomed to deriving personal teaching efficacy from successful
implementation of well-prepared, teacher-centered presentations must be willing to embrace the uncertainty inherent in teaching practices aligned with mathematics-reform (Cooney & Shealy, 1997; Smith, 1996). Furthermore, as teachers transform their classrooms into discourse communities, there are greater opportunities for students to reveal their
understandings and misunderstandings. This circumstance makes it likely that teachers will become more aware of what students do not fully understand (Ball, 1996).
school mathematics, knowledge of how students learn mathematics, and extensive
pedagogical knowledge (Ball et al., 2001). In discussing findings from a study of teachers as they begin to explore multiple solutions for mathematics problems with their students, Silver, Ghousseini, Gosen, Charalambous, and Strawhun (2005) make the following observation:
If teachers lack a sound, flexible knowledge of mathematics and of children's thinking, they may be more inclined toward managing multiple solutions through a ritualized "show-and-tell" practice, which allows them to avoid the complexity of choosing certain solutions, arranging them in a sequence, and connecting them to extract and highlight important mathematical points (p.298).
These researchers found that teachers became more willing and capable of productively incorporating multiple student-generated solutions into their mathematics instruction as they became increasingly aware of particular pedagogical techniques that could be used to make discussion of student solutions work toward their instructional goals. In addition to needing a robust knowledge base for reform-oriented mathematics teaching, teachers must hold beliefs that support an orientation toward mathematics instruction aligned with mathematics-reform.
manner consistent with the principles of reform. It is important for teachers to hold beliefs that support an orientation toward mathematics instruction aligned with mathematics-reform in these types of situations, but it becomes essential when teachers must defend the new ways of teaching.
Teachers who engage in reform-based mathematics practices often find themselves having to explain these new ways of teaching to administrators, parents, and students who are accustomed to different kinds of mathematics instruction (Ball, 1996). Cooney (1985)
describes a first-year teacher who began the school year with a clear vision of a new way to teach mathematics, but resistance from students and pressures to cover the curriculum convinced him to retreat to more traditional practices. Silver et al. (2005) have also found many teachers are resistant to incorporating particular recommendations of reform due to time constraints and perceived conflict between implementing reform-based practices and being able to cover the curriculum.
In all schools, teachers attempting to adopt reform-based mathematics pedagogy face an array of challenges. Teachers in urban schools face additional challenges. Next, I will briefly review challenges associated with urbanicity.
The Challenge of Urbanicity
experiences, and student outcomes. Trends indicate that the proportion of urban school students who are living in poverty or have difficulty speaking English is on the rise. Urban students are more likely to be exposed to safety and health risks, and they are less likely to have the family structure, economic security, and stability that are most associated with desirable education outcomes. Urban school students are more likely than their suburban and rural school counterparts to have changed schools frequently, and they are more likely to be absent from school. Compared to their suburban and rural counterparts, teachers in urban schools report having fewer resources, less control over the curriculum, and higher levels of student behavior problems.
Lippman et al. (1996) found many of these trends to be magnified in urban schools that are also characterized by high concentrations of students in poverty. While the
challenges presented by urbanicity and poverty are sure to place additional demands on teachers, this report also challenges the perception that urban schools with the highest
poverty concentrations are always much worse off than other schools. This study reports high variation among schools, suggesting that some urban schools are successfully meeting the challenges that face them.
In a large-scale study of successful high-poverty schools, Knapp and his colleagues (Knapp, 1995b) found that the more teachers focused on teaching mathematics in ways that emphasize conceptual understanding, the more likely students were to demonstrate
promote mathematics-reforms that emphasize student thinking in urban, high-poverty schools.
Many factors appear to influence the ways teachers approach mathematics instruction in their classrooms. As the act of teaching has been recognized as complex, researchers have also begun to better understand the role teachers’ cognitions play in instructional decision-making and teacher behavior. As was touched on previously, researchers have come to believe that a teacher’s beliefs and knowledge strongly influence how she understands the recommendations of mathematics-reform (Spillane & Zeuli, 1999) and how she engages in mathematics instruction in her classroom (Calderhead, 1996; Ernest, 1989; Fennema & Franke, 1992; Schoenfeld, 1998, 2000; Thompson, 1992). Fennema and Franke (1992) suggest that the relationship between beliefs, knowledge, and practice is an interactive one. They posit that, “Within a given context, teachers' knowledge of content interacts with
knowledge of pedagogy and students' cognitions and combines with beliefs to create a unique set of knowledge that drives classroom behavior” (p.162). The sections that follow will elaborate more fully on how teachers’ knowledge and beliefs are thought to influence mathematics teaching.
Knowledge for Teaching Mathematics
matter knowledge, calling it the missing paradigm in research on teaching. Although there is not yet a consensus on the knowledge needed to teach mathematics (Ball et al., 2001), much headway has been made in exploring the various kinds of knowledge that support teaching (Borko & Putnam, 1996; Ernest, 1989; Fennema & Franke, 1992; Hill et al., 2007; Shulman, 1986, 1987) as well as how that knowledge is mobilized in the act of teaching (Leinhardt, 1993; Schoenfeld, 1998, 2000; Sherin, 2002b).
Borko and Putnam (1996) identify three main categories of knowledge that support teaching: 1) general pedagogical knowledge, 2) subject-matter knowledge, and 3)
pedagogical content knowledge. General pedagogical knowledge refers to important knowledge about teaching, learners, and learning that transcends particular subject matter domains. This includes knowledge about effective strategies for planning, classroom
routines, conducting lessons, and classroom management as well as general knowledge about how children think and learn. Subject-matter knowledge refers to knowledge of the discipline of mathematics that is not unique to teaching. This includes knowledge of the important facts, concepts, and procedures as well as knowledge of the concepts underlying the procedures and relationships between important concepts and mathematical ideas.
The third category of knowledge identified by Borko and Putnam, pedagogical content knowledge, refers to the unique set of subject matter knowledge used in teaching. As first described by Shulman (1986), pedagogical content knowledge includes, “the ways of representing and formulating the subject that make it comprehensible to others,” and, “…an understanding of what makes learning of specific topics easy or difficult: the conceptions and preconceptions that students of different ages and backgrounds bring with them to the
knowledge needed by teachers of the instructional strategies that will help students to
reorganize and deepen their understanding of subject matter. Grossman (1990) elaborated on Shulman’s initial description of pedagogical content knowledge to include knowledge of curriculum and curricular materials. Other researchers have highlighted the importance of understanding student cognitions in particular mathematics domains (Fennema & Franke, 1992).
Thompson and Thompson’s (1994) description of one teacher’s work with an individual student on understanding the concept of rates highlights the importance of pedagogical content knowledge. In a one-on-one tutoring session, the teacher focused on procedural, algorithmic aspects of problems posed, assuming that his student’s correct answers were evidence that she understood the concepts underlying the procedures. When the student became stuck, the teacher was unable to recognize the source of her difficulty. In this case, the teacher held a strong personal understanding of the concept he was teaching, but his understanding was not flexible enough to consider multiple ways of thinking about rates or the trajectory of his student’s mathematical understandings.
their cognitive attention on managing a class discussion of the varied mathematical solutions provided by students, they sometimes have difficulty staying focused on mathematics content goals (Williams & Baxter, 1996). Montero-Sieburth (1989) asserts that the complexities of teaching are compounded in the urban school context:
Today's urban teachers are in a tenuous position. Research evidence has shown that teachers within impoverished urban schools are, at best, so encapsulated and overwhelmed by the demands of their teaching environments that they can barely function. They carry theory around in their heads, but they often do not know how to apply this knowledge in the given context because they are so immersed in practice. Urban teachers in the thick of their routines hardly have the time or energy to reflect on their experiences or their teaching styles (p.337).
It is important to consider the extent to which and how teachers organize their knowledge such that it can be efficiently accessed while teaching.
In the educational psychology literature, it is widely accepted that humans organize their experiences and subsequent knowledge in schemata that help to make sense of their experiences in increasingly efficient ways. In expert-novice studies of teaching, expert teachers have been found to have much more elaborate, interconnected, and flexible
cognitive schemata than novices (Borko & Livingston, 1989; Leinhardt, 1993). Leinhardt and her colleagues (Leinhardt, 1993; Leinhardt, Putnam, Stein, & Baxter, 1991) identify some varieties of schema that teachers use to organize their teaching. These include agendas, routines, and curriculum scripts. Schoenfeld (1998; 2000) describes a similar set of constructs that he calls lesson images and action plans.
An agenda is the teacher’s dynamic “mental notepad” for a lesson (Leinhardt, 1993). Schoenfeld (1998; 2000) discusses a similar construct that he calls a lesson image. He
his or her sense of how the discussion will go” (Schoenfeld, 2000, p.250). The primary function of an agenda or a lesson image is to provide a conceptual roadmap charting the direction of a lesson. It includes the overarching goals and anticipated actions of a lesson and focuses primarily on the non-routine parts of that lesson. In general, experts incorporate more detail into their agendas than novices, and they have a better sense of how a given lesson will play out (Leinhardt, 1993). Expert agendas typically include an image of teacher and student actions, anticipation of student responses, plans for checking student learning at multiple points in the lesson, and subsequent branched plans based on assessment of student understanding. In contrast, novice agendas focus on plans for teacher moves, with little attention to student response, and include few contingency plans. With limited pedagogical content knowledge, especially knowledge of student cognitions, a teacher is limited in her ability to devise an agenda or lesson image that will adequately facilitate a reform-oriented mathematics lesson (Smith, 2000).
Schoenfeld (1998; 2000) identifies action plans as mechanisms for accomplishing goals while teaching. Routines and scripts are two types of action plans that are frequently used. Routines are social scripts that facilitate management and classroom norms (Leinhardt, 1993). Established routines are necessary in teaching because they help to reduce the
teacher’s cognitive load and facilitate focus on non-routine aspects of teaching. Routines can meet a diverse array of goals, from distribution of supplies to establishing norms for
student ideas and strategies while ensuring that her class focused on established mathematics goals. Routines serve an essential role in classroom teaching, but they also can be limiting:
When teachers wish to modify their teaching, they often adopt the large pieces of a new reform (e.g., small group, cooperative teams, problem-centered inquiry, etc.), but they keep the old routines for producing and sharing knowledge. This has two
consequences: First, the new system does not work, and they have management problems; second, the class receives mixed implicit messages. The conflict of routines with philosophies or social organization creates serious difficulties (Leinhardt, 1993, p.18).
As teachers transition to reform-based mathematics pedagogy it is important that they intentionally cultivate new routines that support new goals.
Curriculum scripts (Leinhardt et al., 1991) or scripts (Schoenfeld, 2000) are action plans that detail content-specific scenarios for ways in which segments of instruction will play out. Scripts can be flexible and interactive with spaces for student actions or they can be more rigid, following a specific progression of teacher-determined ideas. These scripts are derived from a conglomeration of subject-matter knowledge and pedagogical content
knowledge that can be easily accessed in the course of a lesson. Sherin (2002b) uses the term content knowledge complexes to describe these interwoven pieces of knowledge. Because of the way these aspects of knowledge are interwoven and accessed together, she suggests that they represent larger elements of accessible teacher knowledge that cannot be characterized as solely subject-matter knowledge or pedagogical content knowledge.
of contingency explanations that they can turn to if needed (Leinhardt, 1993). Carpenter, Fennema, Peterson, & Carey conducted a study (1988) of how first grade teachers use their pedagogical content knowledge of children’s solutions to addition and subtraction word problems during instruction. This study found that most teachers could identify essential elements of problems as well as common student strategies. However, this information did not seem to be organized in ways that supported teachers in making instructional decisions that utilized this knowledge. Given the uncertain nature of classrooms that focus on student thinking, knowledge of alternative explanations and representations that can be accessed at a moment’s notice seems especially important.
It is important to keep in mind that teachers do not always have subject-matter and pedagogical content knowledge to support instruction that builds on student thinking. Lehrer and Franke (1992) compare the way a first-grade teacher, Ms. Jackson, approached
instruction on addition/subtraction concepts to the way she approached instruction on
fractions. Ms. Jackson’s measured knowledge of fractions was much less developed than her knowledge of addition and subtraction, and her methods of instruction were also notably different in these two content areas. While teaching addition and subtraction concepts, the central activity of Ms. Jackson’s class was problem solving using a variety of problem types. Ms. Jackson orchestrated class discussions in which she intentionally elicited a variety of student strategies and supported students in making sense of the mathematical ideas
Warfield (2001) argues that the depth, breadth, and organization of teachers'
mathematical knowledge for teaching influences their abilities to attend to and build on the mathematical thinking of children in their classes. Teachers’ knowledge of children's
thinking and the mathematics they teach influences the extent to which teachers can critically examine their students’ thinking to determine if it is mathematically valid. Additionally, limits in teachers’ knowledge impact teachers’ abilities to pose questions, respond to students’ novel ideas and strategies, and press children to extend their thinking.
It is clear that teaching mathematics in ways that honor student thinking requires greater amounts and varied types of knowledge. Furthermore, this knowledge needs to be organized in a way that makes it useful as teachers plan and implement instruction. The content of a teacher’s knowledge base as well as its organization appears to facilitate and limit the ways in which teachers are capable of supporting and extending student thinking.
Next a brief discussion of knowledge thought to support reform-oriented teaching of multiplication and division will be presented, since these are the mathematics topics of focus in the data for this research. Rather than providing an exhaustive review of the literature pertinent to knowledge for teaching multiplication and division, the section that follows is designed to selectively preview aspects of knowledge for teaching multiplication and
Knowledge for Teaching Multiplication and Division
This discussion of knowledge for teaching multiplication and division begins with an overview of problem types and their relationship to students’ initial problem solving
strategies. Next, students’ progression through increasingly efficient strategies for single-digit multiplication and related division problems will be reviewed, with particular attention to things that influence strategy use. This will be followed by a brief discussion of the array model for multiplication and division. Finally, students’ strategies and learning trajectory for multidigit multiplication will be described.
There are a variety of situations that can be represented by multiplication and
division. These situations can be classified into symmetric and non-symmetric problem types (Greer, 1992). Symmetric problem types include array problems, area problems, and
combinations problems. Non-symmetric problem types include problems involving equal groupings of discrete objects, rate problems, and multiplicative comparison problems. It is important for teachers to have a conception of multiplication and division that includes these various problem types for students to encounter multiplication and division problems that encourage different kinds of thinking and involve different kinds of quantities. For instance, students’ initial development of multiplication and division understanding is supported by problems involving equal groupings of discrete objects. But rate problems, multiplicative comparison problems, and area problems can be extended to rational numbers. Exposure to these additional problem types helps to lay the foundation for multiplication and division of fractions in later grades (Carpenter, Fennema, Franke, Empson, & Levi, 1999).
in a manner that builds on student thinking (Carpenter et al., 1996; Franke et al., 2007). Multiplication and division instruction often begins with equal-grouping problems. This type of problem involves three quantities: a) the number of groups, b) the number of objects in a group, and c) the total number of objects. The quantities known and missing determine the type of equal-grouping problem. Problems in which the total number of objects is unknown (a×b = ?) can be called groups-of multiplication. Problems in which the number of objects
in one group is unknown (a× ? = c or c÷a = ?) are typically called partitive division
problems, corresponding to the social practice of sharing equally. Problems in which the number of groups is unknown (? ×b = c or c÷b = ?) are referred to as measurement division
problems or quotative division problems. An example of each of these problem types is presented in Table 1.
Table 1
Equal-Grouping Problem Types
Problem type Unknown Example
Groups-of multiplication
Total number of objects
Josh has 4 bags with 6 cookies in each bag. How many cookies does Josh have in total?
Partitive division Number in one group
Josh has 24 cookies. He wants to put the same number of cookies in each of 4 bags. How many cookies should Josh put in each bag?
Measurement division Number of groups
Josh is making cookie bags with 6 cookies in a bag. If there are 24 cookies in total, how many bags of 6 cookies can Josh make?
1996). In order to directly model groups-of multiplication problems children make the specified number of groups (4 bags) with the specified number of objects in each group (6 cookies) and then count the total number of objects (24 cookies). For partitive division
problems, children initially represent the specified number of objects (24 cookies) and groups (4 bags) and then use a guess-and-check strategy to determine the numbers of objects
(cookies) that can be divided evenly into each group such that all the objects are used. Later, children develop a more strategic approach to partitive division problems in which they distribute the objects in a systematic manner among the groups. For measurement division problems, children make sets of the number of objects specified for one group (6 cookies) until they have reached the total number of objects (24 cookies). The answer to the
measurement division problem is then found by counting the number of sets or groups formed (4 bags).
As is evident from the direct modeling strategies described, children do not initially understand these problem types in terms of multiplication and division as adults do.
Consequently, research suggests that instruction that builds on student thinking should begin with students’ ways of solving these problems and introduction of concepts and symbolic representation should build on these student-generated solutions (Carpenter et al., 1999). For teachers, it is helpful to have a sense of the learning trajectory students might take as they develop more efficient strategies, including knowledge of the understandings students must acquire to make sense of increasingly sophisticated strategies.
2001). Called unitizing, students are challenged to simultaneously keep track of two counts: the number of objects and the number of groups of objects. There have been many research-based accounts of how children move from direct modeling with objects (to solve equal-grouping problem types) to increasingly efficient number-based strategies (see, e.g.,
Carpenter et al., 1999; Fosnot & Dolk, 2001; Kouba, 1989; Mulligan & Mitchelmore, 1997; Sherin & Fuson, 2005). Verschaffel, Greer, and De Corte (2007) summarize children’s progression through strategies for single-digit multiplication and division as follows:
Generally speaking, children progress from (material-, fingers-, or paper-based) concrete counting-all strategies, through additive-related calculations (repeated adding and additive doubling), pattern-based (e.g., multiplying × 9 as by 10 – 1), and
derived-fact strategies (e.g., deriving 7 × 8 from 7 × 7 = 49) to a final mastery of
learned multiplication products (p.562).
number of objects in each group. However, for partitive division problems, the number of objects in each group is the unknown. The distribution action utilized as children are direct modeling partitive division problems does not translate as easily to additive strategies.
As students begin to move beyond direct modeling strategies, strategy use also varies depending on the numbers involved in a problem and students’ number-specific
computational resources. Drawing on the retrieval-focused literature (Campbell & Graham, 1985), problems with smaller operands (e.g., 2 × 3) are solved by learned product more
quickly than problems with larger operands (e.g., 7 × 8). However, children appear able to
use learned multiplication strategies for problems in which operands are the same (e.g., 6 ×
6) and problems involving 5 as an operand (e.g., 5 × 8) more quickly than is suggested by
their operand-size. Another factor that influences strategy use is the number-specific computational resources students have acquired to operate on the numbers in a problem (Sherin & Fuson, 2005). For instance, students typically learn skip counting sequences for 2, 5, and 10 and use these sequences in problems involving these numbers. However, use of skip counting sequences for problems involving other numbers such as 4 or 7 depend on whether these sequences have been emphasized instructionally.
Implicit or explicit understanding of properties also influences the strategies children use (Fosnot & Dolk, 2001). For example, knowledge of the distributive property allows children to successfully use derived fact strategies, such as solving 8 × 6 by calculating (4 ×
6) + (4 × 6). Knowledge of the commutative property of multiplication (e.g., 5 × 6 = 6 × 5)
allows children to think about the numbers in problems more flexibly. In a problem involving finding the total number of sodas in five six-packs, a student with knowledge of the
problem calls for groups of six. However, it should be noted that some researchers have found that, even when students demonstrate knowledge of the commutative property when dealing with symbols on paper, they do not as readily apply this property to make
calculations easier when numbers appear in contexts (Ambrose, Baek, & Carpenter, 2003). In reform-oriented classrooms, teachers are responsible for orchestrating discussion of students’ strategies for solving multiplication and division problems such that the relationships among strategies and important mathematical ideas, such as the commutative and distributive properties, are illuminated. This involves skillfully posing tasks that lend to use of particular strategies and stimulate discussion of particular ideas.
Some mathematics curriculums, including the reform-based mathematics curriculum utilized by teachers in this study, explicitly introduce arrays to model multiplication and division. An array is a rectangular arrangement of objects in rows and columns such that each row has an equal number of objects (Van de Walle, 2007). Arrays are named by stating the number of rows by the number of objects in a row. (See Figure 1.)
Figure 1. Naming conventions for arrays.
X X X X X X X X X X X X X X X
← 3 ×5 array
3 rows of 5 objects
X X X X X X X X X X X X X X X
← 5 ×3 array
5 rows of 3 objects
problem context suggests an array (Carpenter et al., 1999). Therefore, in introducing the array model, teachers are challenged to organize instruction such that children build
knowledge of arrays through connections to their own, more natural strategies. Additionally, teachers need to be aware that many students initially have difficulty understanding the array structure, particularly understanding how one square can simultaneously be part of a column and a row (Battista, Clements, Arnoff, Battista, & Van den Borrow, 1998). Consequently, teachers using the array representation must be sensitive to students’ developing
understanding of arrays and how students are relating the array structure to problems posed. After children construct meaning of multiplication and division and begin to develop increasingly efficient strategies for working with smaller numbers, instruction turns to a focus on problems involving multidigit calculations. As is the case with calculations with smaller numbers, it is important for teachers to be able to anticipate strategies students will use to solve multidigit problems as well as how students might naturally move through these strategies.
Baek (1998) classifies children's invented solutions to multidigit multiplication problems into four categories1: direct modeling, complete number strategies, partitioning number strategies, and compensating strategies. Direct modeling strategies, the most basic strategy type, entail modeling each group of objects in a multiplication problem with concrete manipulatives or drawings to count the total number of objects. Complete number strategies are based on repeated addition of multiplicands, but do not involve partitioning of the multiplier or multiplicand in any particular way. Doubling strategies that shorten the addition procedure are included in this category. Partitioning strategies involve partitioning
1
the multiplier and/or the multiplicand into smaller numbers so they can be multiplied more easily. Within this category of problem, Baek distinguishes between partitioning into decade and non-decade numbers (e.g., partitioning 16 into 10 and 6 opposed to 8 and 8).
Compensating strategies involve adjusting the multiplier and/or multiplicand up or down based on special characteristics of the number combination to make the calculation easier. Then, after major calculations have been completed, students compensate for the initial adjustments to the numbers. For instance, in solving the problem 4 × 19 = ?, a student might
first find the product of 4 × 20, and then subtract 4 from this product to compensate for the
adjustment to the original problem.
In her research, Baek (1998) found that the students studied progressed through invented multidigit multiplication strategies from direct modeling to complete number to partitioning numbers into non-decade numbers to partitioning numbers into decade numbers. Children’s strategies for solving multidigit multiplication problems varied with their
conceptual knowledge of addition, base-ten knowledge, knowledge of basic multiplication facts and their relationship to multidigit problems, and properties of the four operations. For instance, student use of partitioning and compensating strategies was dependent on students' knowledge of multiplication facts and the distributive property. Development of
understanding of what happens to a number when it is multiplied by a power of ten (e.g., 6 ×
4 = 24, 6 × 40 = 240, 6 × 400 = 2,400) was found to be essential knowledge to using
partitioning strategies involving decade numbers.
certain knowledge does not ensure that she will choose to act on that knowledge. To better understand how teachers choose to use the knowledge they have, I will now turn to a discussion of teachers’ beliefs.
Beliefs and Mathematics Teaching
Beliefs and knowledge are not easily distinguishable. Indeed, people often describe their beliefs in terms of things they “know” (Thompson, 1992). However, beliefs generally refer to suppositions, commitments, and ideologies, while knowledge is viewed as factual propositions and understandings (Calderhead, 1996). Teachers’ beliefs are important because they influence teachers’ perceptions and interpretations of events (Pajares, 1992), and serve as a guiding force in the kinds of actions teacher take (Cooney, Shealy, & Arvold, 1998). Considered in this context, beliefs are the implicit and explicit personal philosophies held by teachers consisting of their conceptions, ideologies and values that shape practice and direct knowledge (Ernest, 1989).
In considering teachers’ beliefs, it is important to distinguish between the content of beliefs and the structure of the beliefs system. Before discussing in greater depth the content of teachers’ beliefs in relation to mathematics teaching, I will briefly review theory relevant to the structure of beliefs and how they are held.
Green (1971) identified three dimensions of the beliefs system:
According to Green’s theory, beliefs are held in interdependent ways, with some beliefs following other beliefs in a hierarchical manner. For example, if a teacher believes that students learn mathematics from constructing their own strategies, a belief that logically follows is that the teacher’s role is to provide opportunities for students to engage in mathematics lessons where they devise their own strategies for solving problems. Green’s theory also suggests that beliefs can be held in isolated clusters, making it possible for persons to hold conflicting beliefs.
Additionally, Green’s theory (1971) suggests that individuals hold beliefs at varying levels of conviction. This is especially important to consider when examining the relationship between teacher beliefs and instructional practice. The complexity of teaching requires teachers to act in situations where multiple, sometimes conflicting, beliefs are activated at once (Aguirre & Speer, 2000). The action a teacher chooses to take is thought to be, in part, a result of the prioritization of the strength of beliefs. More recently, researchers have also begun to consider instances when a teacher’s instructional practice appears to conflict with the teacher’s espoused beliefs as being explained by the teacher’s prioritization of goals (Leatham, 2006; Philipp, 2007; Skott, 2001).
that he believed the two struggling students would provide a significant drain on his attention, limiting his ability to attend to the rest of the class, until they felt confident that they had an answer to the problem. In this case, the teacher’s priorities related to general management of the class and building students’ confidence were deemed of higher priority than the teacher’s goal of aiming to provide support such that students assume responsibility for their own learning.
So far, consideration has been given to how teachers hold beliefs in relation to each other and why teachers’ actions may sometimes appear to contradict certain beliefs. Now consideration will be given to the content of teachers’ mathematics-related beliefs and their implications for mathematics instruction. Calderhead (1996) identifies areas in which teachers are found to hold beliefs relevant to their teaching practice. These include beliefs about subject-matter, learners, and learning as well as beliefs about teaching, the role of the teacher, and teachers’ self-related beliefs. Within the mathematics education literature, there is a focus on the importance of teachers’ beliefs related to the nature of mathematics and mathematics teaching and learning (Ernest, 1989; Franke et al., 1997; Franke et al., 2007; Thompson, 1992).
According to Thompson (1992), “A teacher’s conception of the nature of
finding patterns, and reasoning logically. Sometimes referred to as a problem-solving or inquiry-oriented view, this conception emphasizes mathematics as a way of thinking and devising a growing understanding of the world. At the other end of the spectrum, some teachers view mathematics as a static body of knowledge consisting of a collection of “rules without reasons” (Skemp, 1978, p.9). This view of mathematics emphasizes knowledge of established methods for performing mathematics tasks. Teachers’ beliefs about the nature of mathematics have implications for how they will view and approach mathematics teaching (Lerman, 1983).
A teacher’s beliefs or conception of mathematics teaching includes personal philosophies related to the most desirable goals of mathematics instruction, related
instructional approaches and emphasis, what counts as mathematical activity, and appropriate roles of teachers and students during classroom instruction (Thompson, 1992). Kuhs and Ball (1986, as discussed in Thompson, 1992) identify four ways teachers view mathematics instruction. A learner-focused view concentrates on supporting students’ personal construction of mathematical knowledge and is most related to the problem-solving or inquiry-oriented view of mathematics described earlier. Consistent with a constructivist-orientation to teaching, the teacher’s role is to be a facilitator of student learning by asking probing questions and helping students to uncover misunderstandings and new
understandings. This conception of teaching reflects the emphasis of mathematics-reform and places student thinking at the center.
classroom-focused. The conception described as content-focused with an emphasis on conceptual understanding organizes instruction around the discipline of mathematics instead of student thinking. Mathematics is viewed as a static body of knowledge containing an underlying logic that students must come to understand. In the conception identified as content-focused with an emphasis on performance, there is a focus on mastering rules and procedures to get correct answers, but understanding is de-emphasized. Reflecting an instrumentalist view of mathematics, direct instruction is the dominant teaching style and learners are typically passive receivers of knowledge. The final conception of teaching, described as classroom-focused, de-emphasizes the nature of mathematics and theories of learning. Instead, it focuses on instruction derived from knowledge about effective classrooms.
These distinctions are useful in understanding the differences among teachers’
possible conceptions of teaching mathematics; however, it is important to note that individual teachers are likely to hold a conglomeration of beliefs that cut across multiple models in the Kuhs and Ball framework. Despite the seemingly natural connection between theories of learning and theories of teaching, Thompson (1992) makes the following observation:
Although it seems reasonable to expect a model of mathematics teaching to be somehow related to or derived from some model of mathematics learning, for most teachers it is unlikely that the two have been developed and articulated into a coherent theory of instruction. Rather, conceptions of teaching and learning tend to be eclectic collections of beliefs and views that appear to be more the result of their years of experience in the classroom than of any type of formal or informal study (p.135). That beliefs about learning and teaching are not always linked can be explained by Green’s claim that beliefs are held in isolated clusters (1971). However, certain beliefs do seem central to teaching practices that give student thinking a central role in instruction.
(1996) found that teachers whose mathematics instruction focused on supporting and
building on student thinking held a common set of beliefs. These teachers viewed children as coming to their classrooms with mathematical knowledge and the ability to acquire new knowledge by engaging in problem-solving. Related to this belief, teachers believed that students can learn without direct instruction, problem-solving is central to mathematics instruction, and skills and knowledge are interrelated. The second belief that appeared instrumental in classrooms that promoted student thinking was a view that teaching involves listening carefully to students in order to understand their thinking. Also, these teachers believed that understanding of student thinking should inform instruction.
In contrast, Warfield, Wood, and Lehman (2005) found that teachers who believe that only some children can be autonomous learners of mathematics interpret reform
recommendations in unintended ways and, consequently, give limited attention to
Cooney and Shealy (1997) point out that teaching practices advocated by mathematics-reform make teachers’ classroom experiences more problematic and less
predictable and controllable. They ask teachers to accept uncertainty as an on-going reality of classroom life. This is in sharp contrast to more traditionally held views of mathematics that stress rules and order. This "crossing over" to a relativist stance, in which teachers open up to multiple authorities on mathematical knowledge, is central to realizing fundamental change in the way mathematics is taught. Teachers must come to believe that multiple perspectives and flawed solutions are valuable instructionally.
In research examining impediments to teachers adopting reform-oriented teaching practices, Silver et al. (2005) reveal that many of the teachers they studied expressed concern that incorporating discussion of multiple ways of solving mathematical problems, especially flawed solutions, would lead to confusion for students. In a study comparing teachers’ handling of mistakes in U.S. and Italian classrooms, Santaga (2005) found that U.S. teachers generally avoid the public discussion of flawed solutions in favor of a focus on correct answers. When mistakes arise in class discussion of mathematics, teachers typically aim to correct the errors quickly and move back to a discussion of correct answers. This manner of addressing students’ flawed solutions stands in opposition to a reform-oriented view of students’ mistakes as “springboards for inquiry” (Borasi, 1994). Conceived in this manner, students’ flawed solutions provide ripe opportunities for students to learn through
engagement in genuine problem solving involving analysis of correct and incorrect aspects of solutions in efforts to revise solutions to correct.
mathematics-reform often require teachers to dramatically re-conceptualize the nature of mathematics, teaching, and learning (Cooney & Shealy, 1997; Franke et al., 1997). But even if a teacher holds certain beliefs, it does not mean that she will necessarily act in ways that are consistent with those beliefs. Empson and Junk (2004) report a study in which teachers expressed beliefs about it being a good idea to use student mistakes as opportunities for learning. Despite this professed belief, when asked what they would do in response to a particular teaching scenario, the actions described by several teachers did not reflect the professed belief. Empson and Junk suggest that lack of specific knowledge of children's mathematics may limit teachers' abilities to act on beliefs.
In this example, a hint of the complexity in the relationship of beliefs, knowledge, and practice is visible. Like Empson and Junk, Ernest (1989) suggests that no matter how strongly beliefs are held, necessary knowledge must be on hand to back them up in actions. If the base of knowledge supporting the belief is limited, it is unlikely that actions associated with the particular belief will be realized. Ernest also identifies the social context as an important determinant in the way teachers do their work. He suggests that texts, school norms, expectations from superiors, and external tests yield considerable influence on teachers’ prioritization of goals and subsequent practice, regardless of espoused beliefs.
Beliefs Beyond Mathematics
